Gate-tunable infrared plasmons in electron-doped single-layer antimony

We report on a theoretical study of collective electronic excitations in single-layer antimony crystals (antimonene), a novel two-dimensional semiconductor with strong spin-orbit coupling. Based on a tight-binding model, we consider electron-doped antimonene and demonstrate that the combination of spin-orbit effects with an external bias gives rise to peculiar plasmon excitations in the midinfrared spectral range. These excitations are characterized by low losses and negative dispersion at frequencies effectively tunable by doping and bias voltage. The observed behavior is attributed to the spin splitting of the conduction band, which induces interband resonances, affecting the collective excitations. Our findings open up the possibility to develop plasmonic and optoelectronic devices with high tunability, operating in a technologically relevant spectral range.

Figure 1

Band structure and density of states of SL-Sb calculated (a) in the absence of and (b) in the presence of a vertical bias with $V=1$ eV. In each case, the black horizontal line marks the Fermi energy corresponding to the electron doping with concentrations $n_1={10}^{13}$ and $n_2={10}^{14}{\mathrm{cm}}^{-2}$. (c) Fermi-surface contour for the concentration $n_2$ and bias potential $V=1$ eV, with colors corresponding to the expectation value (in units of $\hslash /2$) of the spin operator projected on the direction perpendicular to $\mathrm{\Gamma }\text{−}M$, $S_{\mathrm{\Gamma }\mathrm{M}}^{\perp }\left(\mathbf{k}\right)$. Blue corresponds to the clockwise direction, while red is for the anticlockwise direction. The direction of the total spin per valley is shown by the black arrows. (d) Schematic representation of SL-Sb embedded in the electric field controlled by the gate voltage.

Figure 2

Plasmon loss function $L(\mathbf{q},\omega )$ in SL-Sb calculated for electron doping concentrations $n={10}^{13}$ and ${10}^{14}{\mathrm{cm}}^{-2}$ (top and bottom panels), and bias voltages $V=0$ and 1 eV (left and right panels). The white line represents boundaries of the particle-hole continuum determined by the poles of the polarization function [Eq. (3)], ${\omega }_0^{\left(\mathbf{q}\right)}=\max \left\{E_n^{(\mathbf{k}+\mathbf{q})}-E_m^{\left(\mathbf{k}\right)}\right\}$.

Figure 3

Solutions of the equation $\text{Re}\left\{\det \left[{\epsilon }_{\sigma {\sigma }^{\prime }}^{\left(\mathbf{q}\right)}\left({\omega }_p\right)\right]\right\}=0$, determining lossless plasmon modes calculated for different bias potentials $V$ (in eV), and for two representative values of electron doping $n$ in SL-Sb. The corresponding region of Landau damping is shown in each case by the same color. For clarity, only the highest (“optical”) plasmon mode is shown at each gate voltage.

Figure 4

Frequency dependence of $\mathrm{Re}\left[{\epsilon }_{\mathrm{eff}}(\omega ,q)\right]$ in SL-Sb calculated for a series of $\mathbf{q}$ points along the $\mathrm{\Gamma }\text{−}M$ path. (a) and (b) correspond to $V=0$ and $V=1$ eV, respectively. In both cases, an electron doping of ${10}^{13}{\mathrm{cm}}^{-2}$ is assumed. The inset shows a zoom-in of the region where $\mathrm{Re}\left[{\epsilon }_{\mathrm{eff}}(\omega ,q)\right]=0$, determining the “optical” plasmon modes. Arrows point to the corresponding solutions for the given set of $\mathbf{q}$ points.